Lax morphisms and limits

Steve Lack – 6 May 1998

If T is a 2-monad on a 2-category K, then the existence of limits in the 2-category T-Alg of strict T-algebras, pseudo T-morphisms, and T-transformations is well-understood. The forgetful 2-functor from T-Alg to K creates flexible limits; these are a large and useful class, containing comma objects, inserters, equifiers, products, cotensor products, and inverters; they do not, on the other hand, contain equalizers or pullbacks. They are formed as the closure of the class of pseudolimits and idempotent-splittings. In this talk I look at the existence of limits in the 2-category T-Alg_l of strict T-algebras, lax T-morphisms, and T-transformations. I show that, although the forgetful 2-functor U_l:T-Alg_l-->K does not create lax limits, it does create oplax limits and idempotent-splittings, and therefore admits products, cotensor products, and certain inserters, equifiers, and comma objects. At the end of the talk I speculated on the possibility that the notion of (horizontal) double limit due to Grandis and Pare could be useful in describing the limits which exist in T-Alg_l. There is a double category associated in a natural way to T-Alg_l, and this double category turns out to admit all horizontal double limits provided that K admits all limits.